scholarly journals Waldschmidt constants for Stanley–Reisner ideals of a class of simplicial complexes

2016 ◽  
Vol 15 (07) ◽  
pp. 1650137 ◽  
Author(s):  
Cristiano Bocci ◽  
Barbara Franci
Keyword(s):  
Simplicial Complexes ◽  
Combinatorial Approach ◽  
Ideal Formula ◽  
Symbolic Powers ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant

We study the symbolic powers of the Stanley–Reisner ideal [Formula: see text] of a bipyramid [Formula: see text] over a [Formula: see text]-gon [Formula: see text]. Using a combinatorial approach, based on analysis of subtrees in [Formula: see text] we compute the Waldschmidt constant of [Formula: see text].

On the configurations of codimension two linear subspaces of ℙN with the Waldschmidt constant less than 2

2020 ◽  
pp. 2150178
Author(s):  
Hassan Haghighi ◽  
Mohammad Mosakhani
Keyword(s):  
Hilbert Function ◽  
Free Resolution ◽  
Minimal Free Resolution ◽  
Codimension Two ◽  
Symbolic Powers ◽  
Point Configurations ◽  
Higher Dimensional ◽  
Algebraic Invariants ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of [Formula: see text]-planes [Formula: see text] in [Formula: see text] with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals [Formula: see text] of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of [Formula: see text], as well as the Hilbert function of [Formula: see text].

scholarly journals Initial sequences and Waldschmidt constants of planar point configurations

2017 ◽  
Vol 27 (06) ◽  
pp. 717-729 ◽  
Author(s):  
Łucja Farnik ◽  
J. Gwoździewicz ◽  
B. Hejmej ◽  
M. Lampa-Baczyńska ◽  
G. Malara ◽  
...  
Keyword(s):  
Planar Point ◽  
Symbolic Powers ◽  
Point Configurations ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant

The purpose of this work is to extend the classification of planar point configurations with low Waldschmidt constants initiated in [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Algebra 220 (2016) 2001–2016] and continued in [M. Mosakhani and H. Haghighi, On the configurations of points in [Formula: see text] with the Waldschmidt constant equal to two, J. Pure Appl. Algebra 220 (2016) 3821–3825] for all values less than [Formula: see text]. As a consequence, we prove a conjecture of Dumnicki, Szemberg and Tutaj-Gasińska concerning initial sequences with low first differences.

All symbolic powers and ordinary powers of the defining ideal of a fat nearly-complete intersection are equal

2019 ◽  
Vol 18 (08) ◽  
pp. 1950142 ◽  
Author(s):  
Hassan Haghighi ◽  
Mohammad Mosakhani
Keyword(s):  
Positive Integer ◽  
Complete Intersection ◽  
Containment Problem ◽  
Ideal Formula ◽  
Symbolic Powers

The purpose of this paper is to study the containment problem for the corresponding ideal [Formula: see text] of a special zero dimensional subscheme [Formula: see text] in [Formula: see text], what we call a fat nearly-complete intersection. We show that for every positive integer [Formula: see text], the equality [Formula: see text] holds.

scholarly journals Invariants of the symbolic powers of edge ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050184
Author(s):  
Bidwan Chakraborty ◽  
Mousumi Mandal
Keyword(s):  
Complete Graph ◽  
Explicit Description ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Single Vertex ◽  
Symbolic Powers ◽  
Odd Cycles ◽  
Waldschmidt Constant

Let [Formula: see text] be a graph and [Formula: see text] be its edge ideal. When [Formula: see text] is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of [Formula: see text] and compute the Waldschmidt constant. When [Formula: see text] is complete graph then we describe the generators of the symbolic powers of [Formula: see text] and compute the Waldschmidt constant and the resurgence of [Formula: see text]. Moreover for complete graph we prove that the Castelnuovo–Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

scholarly journals Back to the Coordinated Attack Problem

2020 ◽  
Vol 30 (10) ◽  
pp. 1089-1113 ◽  
Author(s):  
Emmanuel Godard ◽  
Eloi Perdereau
Keyword(s):  
Simplicial Complexes ◽  
Necessary Condition ◽  
Combinatorial Approach ◽  
Topological Characterization ◽  
Topological Approach ◽  
Consensus Algorithms ◽  
Distributed Computability ◽  
Specific Assumption ◽  
The Relationship ◽  
The Way

AbstractWe consider the well-known Coordinated Attack Problem, where two generals have to decide on a common attack, when their messengers can be captured by the enemy. Informally, this problem represents the difficulties to agree in the presence of communication faults. We consider here only omission faults (loss of message), but contrary to previous studies, we do not to restrict the way messages can be lost, i.e., we make no specific assumption, we use no specific failure metric. In the large subclass of message adversaries where the double simultaneous omission can never happen, we characterize which ones are obstructions for the Coordinated Attack Problem. We give two proofs of this result. One is combinatorial and uses the classical bivalency technique for the necessary condition. The second is topological and uses simplicial complexes to prove the necessary condition. We also present two different Consensus algorithms that are combinatorial (resp. topological) in essence. Finally, we analyze the two proofs and illustrate the relationship between the combinatorial approach and the topological approach in the very general case of message adversaries. We show that the topological characterization gives a clearer explanation of why some message adversaries are obstructions or not. This result is a convincing illustration of the power of topological tools for distributed computability.

scholarly journals Symbolic powers in weighted oriented graphs

2021 ◽  
pp. 1-17
Author(s):  
Mousumi Mandal ◽  
Dipak Kumar Pradhan
Keyword(s):  
Oriented Graph ◽  
Explicit Description ◽  
Oriented Graphs ◽  
Edge Ideals ◽  
Underlying Graph ◽  
Symbolic Powers ◽  
Odd Cycle ◽  
Waldschmidt Constant

Let [Formula: see text] be a weighted oriented graph with the underlying graph [Formula: see text] when vertices with non-trivial weights are sinks and [Formula: see text] be the edge ideals corresponding to [Formula: see text] and [Formula: see text] respectively. We give an explicit description of the symbolic powers of [Formula: see text] using the concept of strong vertex covers. We show that the ordinary and symbolic powers of [Formula: see text] and [Formula: see text] behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of [Formula: see text] for certain classes of weighted oriented graphs. When [Formula: see text] is a weighted oriented odd cycle, we compute [Formula: see text] and prove [Formula: see text] and show that equality holds when there is only one vertex with non-trivial weight.

scholarly journals Negative Curves on Symmetric Blowups of the Projective Plane, Resurgences, and Waldschmidt Constants

2018 ◽  
Vol 2019 (24) ◽  
pp. 7459-7514 ◽  
Author(s):  
Thomas Bauer ◽  
Sandra Di Rocco ◽  
Brian Harbourne ◽  
Jack Huizenga ◽  
Alexandra Seceleanu ◽  
...  
Keyword(s):  
Projective Plane ◽  
Singular Points ◽  
Invariant Curves ◽  
Reflection Groups ◽  
Complex Reflection Groups ◽  
Blowing Up ◽  
Configurations Of Lines ◽  
Waldschmidt Constants ◽  
Waldschmidt Constant ◽  
The Ideal

Abstract The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least 3. In this paper we study the surface X obtained by blowing up $\mathbb{P}^{2}$ in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal; ideals with this property are seemingly quite rare. The resurgence and asymptotic resurgence are invariants which were introduced to measure such failures of containment. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

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